Universal inhomogeneous magnetic-field response in the normal state of cuprate high-$T_c$ superconductors

We report the results of a muon spin rotation ($\mu$SR) study of the bulk of Bi${}_{2+x}$Sr${}_{2-x}$CaCu${}_2$O${}_{8+\delta }$, as well as pure and Ca-doped YBa${}_2$Cu${}_3$O${}_y$, which together with prior measurements reveal a universal inhomogeneous magnetic-field response of hole-doped cuprates extending to temperatures far above the critical temperature $T_c$. The primary features of our data are incompatible with the spatially inhomogeneous response being dominated by known charge-density-wave and spin-density-wave orders. Instead, the normal-state inhomogeneous line broadening is found to scale with the maximum value $T_c^{\max }$ for each cuprate family, indicating it is controlled by the same energy scale as $T_c$. Since the degree of chemical disorder varies widely among the cuprates we have measured, the observed scaling constitutes evidence for an intrinsic electronic tendency toward inhomogeneity above $T_c$.

Figure 1

Schematic of the TF-$\mu$SR experimental arrangement. The ${\mu }^{+}$ beam passes through a thin plastic scintillator muon detector with its initial muon-spin polarization P(0) transverse to the direction of the applied magnetic field H, which creates a start pulse for an electronic clock. The muons are subsequently implanted one by one in the sample, where they come to rest and Larmor precess in the local magnetic field B. The time evolution of the muon spin polarization $P\left(t\right)$, which is affected by both static and fluctuating internal magnetic fields, is monitored via the detection of the decay positrons (e${}^{+}$). The detection of a decay positron stops the electronic clock, and the corresponding elapsed-time bin of the positron-detector histogram is incremented. The positron detectors are arranged in pairs on opposite sides of the sample. In our experiments, two pairs of positron detectors surrounding the sample (left and right and up and down) were used. In addition, the sample was mounted directly on a veto detector, used to eliminate muons that missed the sample from contributing to the TF-$\mu$SR signal. The sample covered approximately 40$\%$ to 60$\%$ of the veto detector. (The lower figure is a depiction of the counter arrangement facing the ${\mu }^{+}$ beam). With the exception of the incoming muon detector, all detectors were contained with the sample inside a helium-gas flow cryostat. The TF-$\mu$SR asymmetry spectrum $A\left(t\right)=a_{0}P\left(t\right)$ is formed by combining the accumulated histograms of opposing positron detectors.

Figure 2

Representative TF-$\mu$SR signals. (a) TF-$\mu$SR asymmetry spectrum for optimally-doped ($p=0.165$) YBCO at $H=7$ T and $T=2.7$ K. The two signals, which differ in phase by $\sim {90}^{\circ }$, come from the two pairs of opposing positron detectors shown in Fig. 1 (i.e. Up and Down, and Left and Right). The solid black curves through the data points are fits described in the main text. (b) Same as (a), but for $T=120$ K. (c), (d) Fourier transforms (with Gaussian apodization) of the asymmetry spectra, which provide visual depictions of the internal magnetic field distribution $n\left(B\right)$ sensed by the muon ensemble. The frequency (horizontal axis) is related to the local internal magnetic field via the relation $f=({\gamma }_{\mu }/2\pi )B$.

Figure 3

(a) Temperature dependence of $\Lambda$ at $H=7$ T (solid circles) for BSCCO. (b) Blow-up of the high-temperature data from (a). Also shown are data for the $p=0.094$ sample at $H=0.5$ T (open purple triangles). The arrows indicate the zero-field values of $T_c$ determined by bulk magnetic susceptibility. Representative results for (c) pure and (d) Ca-doped YBCO for $H=7$ T. The data at $p=0.103$ are from Ref. 18. High-temperature behavior of $\Lambda$ (solid circles) for optimally doped (e) BSCCO, (f) YBCO, and (g) LSCO. Also shown is the bulk magnetic susceptibility (light blue curves) for the BSCCO and LSCO samples for $H=7$ T applied parallel to the $c$ axis.

Figure 4

(a) Doping dependence of $\Lambda$ for $H=7$ T and $T=0.11T_c^{\max }$, where $T_c^{\max }=90$, 94.1, and 38 K for BSCCO, YBCO, and LSCO, respectively. The YBCO data for $p\le 0.141$ are from Ref. 18. For visual purposes the relaxation rate for BSCCO has been multiplied by a factor of 2. (b) Results for $T=1.3T_c^{\max }$, where $\Lambda$ is a pure exponential relaxation rate. The BSCCO (blue diamonds) and YBCO (green circles) data are for $T=120$ K, and the LSCO data (red triangles) are for $T=50$ K. Unlike in (a), the BSCCO data are not rescaled. The dashed line is a linear fit of the LSCO data for $p\ge 0.19$, which describes the Curie-like contribution.

Figure 5

Temperature dependence of the bulk dc magnetic susceptibility of the $p=0.094$, 0.16, 0.186, and 0.197 BSCCO single crystals for an external magnetic field $H=7$ T applied parallel to the $c$ axis.

Figure 6

Hole-doping dependence of the exponential relaxation rate $\Lambda$ for LSCO from fits to Eq. (2) (solid red circles). The solid black line in each panel is the best fit of a straight line to the data at $p\ge 0.19$. Subtracting the straight line fit from $\Lambda$ yields the open red circles.

Figure 7

Doping dependence of $\Lambda$ for $T=1.1$, 1.3, and 1.7 $T_c^{\max }$. The $p$-linear-dependent contribution to the data for LSCO has been subtracted, as described in the main text. The solid curves are guides to the eye.

Figure 8

(a) Contour bar graph of the variation of $\Lambda$ with temperature and hole doping in pure and Ca-doped YBCO for $H=7$ T, achieved by interpolation of the $\Lambda$ vs $T$ data sets. The black connected circles indicate the arbitrary value $\Lambda =0.02$ $\mu$s${}^{-1}$. Also shown are the onset temperature $T_{\mathrm{SCF}}$ for SCFs at $p=0.12$ inferred from Nernst-effect measurements and $T^{*}$ (solid red line) extrapolated to higher doping (dashed red line) from Ref. 2. (b) Similar schematic phase diagram for BSCCO, where $T_{\mathrm{dia}}$ is the temperature at which torque magnetometry measurements detect the onset of diamagnetism[11] and $T^{*}$ is from Ref. 40. The $T_{p,\max }$ curve indicates the temperature above which STM measurements[10] on BSCCO find less than 10$\%$ of the sample containing nanometer-sized regions with pairing gaps.